Abstract group 16934047920.b: $\PGL(3,19)$

• Label: 16934047920.b
• Order: \( 2^{4} \cdot 3^{5} \cdot 5 \cdot 19^{3} \cdot 127 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \cdot 127 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{5} \cdot 5 \cdot 19^{3} \cdot 127 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $381$
• Trans deg.: $381$
• Rank: $2$

Group information

Description:	$\PGL(3,19)$
Order:	\(16934047920\)\(\medspace = 2^{4} \cdot 3^{5} \cdot 5 \cdot 19^{3} \cdot 127 \)
Exponent:	\(868680\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \cdot 127 \)
Automorphism group:	Group of order \(33868095840\)\(\medspace = 2^{5} \cdot 3^{5} \cdot 5 \cdot 19^{3} \cdot 127 \)
Composition factors:	$C_3$, $\PSL(3,19)$
Derived length:	$1$

This group is nonabelian, almost simple, and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	15	18	19	20	24	30	36	38	40	45	57	60	72	90	114	120	127	171	180	342	360	381
Elements	1	137541	47327822	47039022	94078044	157071822	94078044	471215466	94078044	94078044	188156088	1725589386	47045880	188156088	188156088	188156088	282234132	49514760	376312176	564468264	99029520	376312176	564468264	564468264	99029520	752624352	1866745440	297088560	1128936528	297088560	2257873056	3733490880	16934047920
Conjugacy classes	1	1	5	1	2	5	2	15	2	2	4	39	2	4	4	4	6	1	8	12	2	8	12	12	2	16	42	6	24	6	48	84	382
Divisions	1	1	3	1	1	3	1	3	1	1	1	7	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	45
Autjugacy classes	1	1	3	1	2	3	1	9	2	1	2	21	2	4	2	2	3	1	4	6	1	4	6	6	1	8	21	3	12	3	24	42	202

Dimension	1	2	380	381	760	762	2286	6480	6858	6859	7239	7620	13716	13718	14478	15240	22860	27432	41148	43434	45720	54864	82296	109728	164592	272160	329184	544320
Irr. complex chars.	3	0	3	15	0	0	0	126	171	3	15	46	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	382
Irr. rational chars.	1	1	1	1	1	1	2	0	1	1	1	2	4	1	1	1	2	4	1	2	6	2	3	1	1	1	1	1	45

Minimal presentations

Permutation degree:	$381$
Transitive degree:	$381$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	380	380	380
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PGL(3,19)$, $\PGammaL(3,19)$
Permutation group:	Degree $381$ $\langle(4,218,14,308,213,146,113,202,242,83,376,52,87,29,340,105,262,66)(6,170,183,274,164,171,221,371,252,50,198,173,282,256,235,128,96,315) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$\PSL(3,19)$ . $C_3$				more information

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\GL(3,19)$.

Homology

Abelianization:	$C_{3} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 9820525962 subgroups in 719 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\PGL(3,19)$
Commutator:	$G' \simeq$ $\PSL(3,19)$	$G/G' \simeq$ $C_3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\PGL(3,19)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\PGL(3,19)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\PGL(3,19)$
Socle:	$\operatorname{soc} \simeq$ $\PSL(3,19)$	$G/\operatorname{soc} \simeq$ $C_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9^2:C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $\He_{19}$
127-Sylow subgroup:	$P_{ 127 } \simeq$ $C_{127}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 16934047920: $\PGL(3,19)$

Order 5644682640: $\PSL(3,19)$

Order 44446320: $C_{19}^2.\GL(2,19)$ x 2

Order 22223160: $C_{19}^2.C_{18}.\PSL(2,19)$ x 2

Order 14815440: $C_{19}^2.C_6.\PSL(2,19).C_2$ x 2

Order 7407720: $C_{19}^2.C_6.\PSL(2,19)$ x 2

Order 4938480: $C_{19}^2.C_2.\PSL(2,19).C_2$ x 2

Order 2469240: $C_{19}^2.\SL(2,19)$ x 2

Order 2222316: $\He_{19}.C_{18}^2$

Order 1111158: $\He_{19}.C_9^2.C_2$ x 3

Order 740772: $\He_{19}.C_3.C_6^2$ x 4

Order 555579: $\He_{19}:C_9^2$

Order 389880: $C_{19}^2.C_{18}.A_5$ x 2

Order 370386: $\He_{19}:(C_3\times C_{18})$ x 12

Order 259920: $C_{19}^2.C_{60}.C_6.C_2$ x 2

Order 246924: $\He_{19}:(C_2\times C_{18})$ x 12, $\He_{19}:C_6^2$

Order 233928: $F_{19}\wr C_2$ x 2

Order 185193: $\He_{19}:(C_3\times C_9)$ x 4

Order 155952: $C_{19}^2.Q_8.C_3.C_{18}$ x 2

Order 129960: $C_{19}^2.C_{90}.C_2^2$ x 4, $C_{19}^2.C_6.A_5$ x 2, $F_{361}$ x 2

Order 123462: $\He_{19}:C_{18}$ x 36, $\He_{19}:(C_3\times C_6)$ x 3

Order 123120: $\GL(2,19)$

Order 116964: $C_{19}^2.C_{18}^2$ x 2, $C_{19}^2:(C_9\times D_{18})$ x 2, $C_{19}^2:C_9:C_{36}$ x 2

Order 86640: $C_{19}^2.C_{60}.C_2^2$ x 2

Order 82308: $\He_{19}:(C_2\times C_6)$ x 4

Order 77976: $D_{19}^2:(S_3\times C_9)$ x 2, $D_{19}^2:(C_3\times D_9)$ x 2, $C_{19}^2.Q_8.C_9.C_3$, $C_{19}^2.Q_8.C_3^2.C_3$

Order 64980: $C_{19}^2:(C_{18}\times D_5)$ x 2, $C_{19}^2:(C_9\times C_5:C_4)$ x 2, $C_{19}^2:C_{180}$ x 2

Order 61731: $\He_{19}:C_9$ x 12, $\He_{19}:C_3^2$

Order 61560: $C_9\times \SL(2,19)$

Order 58482: $C_{19}^2:(C_9\times C_{18})$ x 4, $C_{19}^2:(C_9\times D_9)$ x 2

Order 51984: $C_{19}^2.Q_8.C_3.C_6$ x 2, $C_{19}^2.C_{36}.C_2^2$ x 2

Order 43320: $C_{19}^2:(C_3\times D_{20})$ x 2, $C_{19}^2:(C_3\times C_5:Q_8)$ x 2, $C_{19}^2:C_{120}$ x 2, $C_{19}^2:\SL(2,5)$ x 2

Order 41154: $\He_{19}:C_6$ x 12

Order 41040: $\SL(2,19):C_6$

Order 38988: $C_{19}^2:(C_6\times C_{18})$ x 6, $C_{19}^2:(S_3\times C_{18})$ x 2, $C_{19}^2:C_3:C_{36}$ x 2, $C_{19}^2:C_9:C_{12}$ x 2, $C_{19}^2:(C_3\times D_{18})$ x 2

Order 32490: $C_{19}^2:(C_9\times D_5)$ x 2, $C_{19}^2:C_{90}$ x 2

Order 29241: $C_{19}^2:C_9^2$ x 2

Order 28880: $C_{19}^2:(C_8:D_5)$ x 2

Order 27436: $\He_{19}:C_2^2$

Order 25992: $C_{19}^2:(Q_8:C_9)$ x 2, $C_{19}^2:(C_9\times Q_8)$ x 2, $C_{19}^2:(C_3\times \SL(2,3))$ x 2, $C_{19}^2:C_{72}$ x 2, $D_{19}^2:C_{18}$ x 2, $D_{19}^2:(C_3\times S_3)$ x 2, $D_{19}^2:D_9$ x 2

Order 21660: $C_{19}^2:(C_6\times D_5)$ x 2, $C_{19}^2:(C_3\times C_5:C_4)$ x 2, $C_{19}^2:C_{60}$ x 2

Order 20577: $\He_{19}:C_3$ x 2, $\He_{19}:C_3$, $\He_{19}:C_3$

Order 20520: $C_3\times \SL(2,19)$

Order 19494: $C_{19}^2:(C_3\times C_{18})$ x 14, $C_{19}^2:(S_3\times C_9)$ x 2, $C_{19}^2:(C_3\times D_9)$ x 2

Order 17328: $C_{19}^2:\GL(2,3)$ x 2, $C_{19}^2:(C_3\times \SD_{16})$ x 2

Order 16245: $C_{19}^2:C_{45}$ x 2

Order 14440: $C_{19}^2:D_{20}$ x 2, $C_{19}^2:(C_5:Q_8)$ x 2, $C_{19}^2:C_{40}$ x 2

Order 13718: $\He_{19}:C_2$ x 2, $\He_{19}:C_2$

Order 13680: $\SL(2,19):C_2$

Order 12996: $C_{19}^2:(C_2\times C_{18})$ x 12, $C_{19}^2:C_6^2$ x 2, $C_{19}^2:D_{18}$ x 2, $C_{19}^2:C_{36}$ x 2, $D_{19}:F_{19}$ x 2, $C_{19}^2:(C_6\times S_3)$ x 2, $C_{19}^2:C_3:C_{12}$ x 2, $C_{19}^2:C_9:C_4$ x 2

Order 10830: $C_{19}^2:(C_3\times D_5)$ x 2, $C_{19}^2:C_{30}$ x 2

Order 9747: $C_{19}^2:(C_3\times C_9)$ x 6

Order 8664: $C_{19}^2:\SL(2,3)$ x 4, $C_{19}^2:(C_3\times Q_8)$ x 2, $D_{19}^2:C_6$ x 2, $D_{19}^2:S_3$ x 2, $C_{19}^2:C_{24}$ x 2

Order 7220: $C_{19}^2:D_{10}$ x 2, $C_{19}^2:(C_5:C_4)$ x 2, $C_{19}^2:C_{20}$ x 2

Order 6859: $\He_{19}$

Order 6840: $\PGL(2,19)$, $\SL(2,19)$

Order 6498: $C_{19}^2:C_{18}$ x 35, $C_{19}^2:(C_3\times C_6)$ x 4, $C_{19}:F_{19}$ x 2, $C_{19}^2:(C_3\times S_3)$ x 2, $C_{19}^2:D_9$ x 2, $C_{19}:F_{19}$ x 2

Order 6156: $C_{18}\times F_{19}$

Order 5776: $C_{19}^2:\SD_{16}$ x 2

Order 5415: $C_{19}^2:C_{15}$ x 2

Order 4332: $C_{19}^2:(C_2\times C_6)$ x 4, $D_{19}^2:C_3$ x 2, $C_{19}^2:C_3:C_4$ x 2, $C_{19}^2:D_6$ x 2, $C_{19}^2:C_{12}$ x 2

Order 3610: $C_{19}^2:C_{10}$ x 2, $C_{19}^2:D_5$ x 2

Order 3420: $\PSL(2,19)$

Order 3249: $C_{19}^2:C_9$ x 3, $C_{19}^2:C_9$ x 2, $(C_{19}:C_3)^2$ x 2, $C_{19}^2:C_9$ x 2, $C_{19}^2:C_9$ x 2, $C_{19}^2:C_9$ x 2, $C_{19}^2:C_9$ x 2, $C_{19}\times C_{19}:C_3.C_3$ x 2

Order 3078: $C_9\times F_{19}$ x 2, $C_{171}:C_{18}$

Order 2888: $C_{19}^2:Q_8$ x 2, $D_{19}\wr C_2$ x 2, $C_{19}^2:C_8$ x 2

Order 2166: $C_{19}^2:C_6$ x 3, $C_{19}^2:C_6$ x 2, $C_{19}^2:C_6$ x 2, $C_{19}^2:C_6$ x 2, $C_{19}\times D_{19}:C_3$ x 2, $D_{19}\times C_{19}:C_3$ x 2, $C_{19}^2:S_3$ x 2, $C_{19}^2:C_6$ x 2

Order 2052: $C_6\times F_{19}$ x 3, $C_{342}:C_6$

Order 1944: $C_9^2:S_4$

Order 1805: $C_{19}^2:C_5$ x 2

Order 1539: $C_{171}:C_9$

Order 1444: $D_{19}^2$ x 2, $C_{19}^2:C_4$ x 2

Order 1143: $C_{381}:C_3$

Order 1083: $C_{19}^2:C_3$ x 3, $C_{19}^2:C_3$ x 2, $C_{19}:C_{57}$ x 2

Order 1080: $C_9\times \SL(2,5)$

Order 1026: $C_3\times F_{19}$ x 6, $C_{114}:C_9$ x 3, $C_{171}:C_6$ x 2, $C_{171}:C_6$

Order 972: $C_{18}^2:C_3$

Order 722: $C_{19}\times D_{19}$ x 3, $C_{19}:D_{19}$ x 2

Order 720: $C_{360}:C_2$

Order 684: $C_2\times F_{19}$ x 9, $C_{38}:C_{18}$ x 2, $C_{114}:C_6$, $C_9\times D_{38}$

Order 648: $(C_3\times C_9):S_4$, $C_{18}\wr C_2$

Order 513: $C_{57}:C_9$ x 3, $C_{171}:C_3$

Order 486: $C_9^2:S_3$

Order 432: $C_9\times \GL(2,3)$

Order 381: $C_{127}:C_3$ x 3, $C_{381}$

Order 361: $C_{19}^2$ x 3

Order 360: $C_3\times \SL(2,5)$, $C_{360}$, $C_9\times D_{20}$, $C_{45}:Q_8$, $A_6$

Order 342: $F_{19}$ x 19, $C_{19}:C_{18}$ x 9, $C_{19}:C_{18}$ x 4, $C_{57}:C_6$ x 2, $C_9\times D_{19}$ x 2, $C_{19}:C_{18}$ x 2, $C_{342}$, $C_{114}:C_3$

Order 324: $C_6^2.C_3^2$ x 2, $C_{18}^2$, $C_9\times D_{18}$, $C_9:C_{36}$

Order 243: $C_9^2:C_3$

Order 240: $C_{120}:C_2$

Order 228: $C_{38}:C_6$ x 3, $C_3\times D_{38}$

Order 216: $C_3^2:S_4$, $D_6:C_{18}$, $D_{18}:C_6$, $C_9\times \SL(2,3)$, $\PU(3,2)$

Order 180: $C_9\times D_{10}$, $C_{180}$, $C_5:C_{36}$

Order 171: $C_{19}:C_9$ x 10, $C_{19}:C_9$ x 2, $C_{57}:C_3$, $C_{171}$

Order 162: $C_9\times D_9$, $C_9\times C_{18}$, $\He_3:S_3$

Order 144: $C_9\times \SD_{16}$, $C_3\times \GL(2,3)$

Order 127: $C_{127}$

Order 120: $\SL(2,5)$, $C_{120}$, $C_3\times D_{20}$, $C_{15}:Q_8$

Order 114: $C_{19}:C_6$ x 7, $C_{19}:C_6$ x 3, $C_3\times D_{19}$ x 2, $C_{114}$

Order 108: $C_6\times C_{18}$ x 2, $C_3^2:A_4$ x 2, $C_3:C_{36}$, $C_9:C_{12}$, $S_3\times C_{18}$, $C_3\times D_{18}$

Order 90: $C_{90}$, $C_9\times D_5$

Order 81: $\He_3:C_3$ x 2, $C_9^2$

Order 80: $C_{40}:C_2$

Order 76: $D_{38}$

Order 72: $C_9:D_4$, $C_3:S_4$, $\PSU(3,2)$, $C_6\wr C_2$, $Q_8:C_9$, $C_3\times \SL(2,3)$, $C_{72}$, $Q_8\times C_9$, $D_4\times C_9$

Order 60: $A_5$, $C_{60}$, $C_5:C_{12}$, $C_3\times D_{10}$

Order 57: $C_{19}:C_3$ x 4, $C_{57}$

Order 54: $C_3\times C_{18}$ x 3, $C_3^2:C_6$, $S_3\times C_9$, $C_3\times D_9$

Order 48: $\GL(2,3)$, $C_3\times \SD_{16}$

Order 45: $C_{45}$

Order 40: $D_{20}$, $C_5:Q_8$, $C_{40}$

Order 38: $D_{19}$ x 3, $C_{38}$

Order 36: $C_2\times C_{18}$ x 3, $C_3\times A_4$ x 2, $C_3^2:C_4$, $C_3:C_{12}$, $D_{18}$, $C_{36}$, $C_6^2$, $C_6\times S_3$, $C_9:C_4$

Order 30: $C_{30}$, $C_3\times D_5$

Order 27: $\He_3$ x 2, $C_3\times C_9$ x 2

Order 24: $\SL(2,3)$ x 2, $C_3:D_4$, $C_{24}$, $S_4$, $C_3\times Q_8$, $C_3\times D_4$

Order 20: $D_{10}$, $C_{20}$, $C_5:C_4$

Order 19: $C_{19}$ x 2

Order 18: $C_{18}$ x 7, $C_3\times C_6$, $C_3:S_3$, $C_3\times S_3$, $D_9$

Order 16: $\SD_{16}$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$ x 2, $A_4$ x 2, $D_6$, $C_{12}$, $C_3:C_4$

Order 10: $C_{10}$, $D_5$

Order 9: $C_3^2$ x 3, $C_9$ x 3

Order 8: $Q_8$, $D_4$, $C_8$

Order 6: $C_6$ x 3, $S_3$

Order 5: $C_5$

Order 4: $C_2^2$, $C_4$

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 16934047920: $\PGL(3,19)$

Order 5644682640: $\PSL(3,19)$

Order 44446320: $C_{19}^2.\GL(2,19)$

Order 2222316: $\He_{19}.C_{18}^2$

Order 555579: $\He_{19}:C_9^2$

Order 123120: $\GL(2,19)$

Order 61560: $C_9\times \SL(2,19)$

Order 41040: $\SL(2,19):C_6$

Order 27436: $\He_{19}:C_2^2$

Order 20520: $C_3\times \SL(2,19)$

Order 13680: $\SL(2,19):C_2$

Order 6859: $\He_{19}$

Order 6156: $C_{18}\times F_{19}$

Order 3420: $\PSL(2,19)$

Order 1944: $C_9^2:S_4$

Order 1539: $C_{171}:C_9$

Order 1143: $C_{381}:C_3$

Order 1080: $C_9\times \SL(2,5)$

Order 972: $C_{18}^2:C_3$

Order 720: $C_{360}:C_2$

Order 486: $C_9^2:S_3$

Order 432: $C_9\times \GL(2,3)$

Order 243: $C_9^2:C_3$

Order 240: $C_{120}:C_2$

Order 127: $C_{127}$

Order 80: $C_{40}:C_2$

Order 76: $D_{38}$

Order 45: $C_{45}$

Order 16: $\SD_{16}$

Order 15: $C_{15}$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 16934047920: $\PGL(3,19)$ ($C_1$)

Order 5644682640: $\PSL(3,19)$ ($C_3$)

Order 1: $C_1$ ($\PGL(3,19)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 16934047920: $\PGL(3,19)$ ($C_1$)

Order 5644682640: $\PSL(3,19)$ ($C_3$)

Order 1: $C_1$ ($\PGL(3,19)$)

Series

Derived series	$\PGL(3,19)$	$\rhd$	$\PSL(3,19)$
Chief series	$\PGL(3,19)$	$\rhd$	$\PSL(3,19)$	$\rhd$	$C_1$
Lower central series	$\PGL(3,19)$	$\rhd$	$\PSL(3,19)$
Upper central series	$C_1$

Character theory

Complex character table

See the $382 \times 382$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $45 \times 45$ rational character table.